A finding of the maximal saddle-node bifurcation for systems of differential equations

A variational method is presented for directly finding the bifurcation point of nonlinear equations as the saddle-node point of the extended nonlinear Rayleigh quotient. The method is applied for solving an open problem on the existence of a maximal saddle-node bifurcation point for set of positive solutions of system equations with convex-concave type nonlinearity.